SLIDE 1 Some aspects of stochastic differential equations driven by fractional Brownian motions
Fabrice Baudoin
Purdue University
Based on joint works with L. Coutin, M. Hairer, C. Ouyang and
SLIDE 2 Motivation
The motivation of the talk is to present some results concerning solutions of stochastic differential equations on Rn X x
t = x +
t V0(X x
s )ds + d
t Vi(X x
s )dBi s
(1) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d-dimensional fractional Brownian motion.
SLIDE 3 Motivation
The motivation of the talk is to present some results concerning solutions of stochastic differential equations on Rn X x
t = x +
t V0(X x
s )ds + d
t Vi(X x
s )dBi s
(1) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d-dimensional fractional Brownian motion. More precisely, we shall be interested in the properties of the distribution of X x
t :
Existence of a smooth density;
SLIDE 4 Motivation
The motivation of the talk is to present some results concerning solutions of stochastic differential equations on Rn X x
t = x +
t V0(X x
s )ds + d
t Vi(X x
s )dBi s
(1) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d-dimensional fractional Brownian motion. More precisely, we shall be interested in the properties of the distribution of X x
t :
Existence of a smooth density; Small-time asymptotics;
SLIDE 5 Motivation
The motivation of the talk is to present some results concerning solutions of stochastic differential equations on Rn X x
t = x +
t V0(X x
s )ds + d
t Vi(X x
s )dBi s
(1) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d-dimensional fractional Brownian motion. More precisely, we shall be interested in the properties of the distribution of X x
t :
Existence of a smooth density; Small-time asymptotics; Smoothing properties of the operator Ptf (x) = E(f (X x
t ));
SLIDE 6 Motivation
The motivation of the talk is to present some results concerning solutions of stochastic differential equations on Rn X x
t = x +
t V0(X x
s )ds + d
t Vi(X x
s )dBi s
(1) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d-dimensional fractional Brownian motion. More precisely, we shall be interested in the properties of the distribution of X x
t :
Existence of a smooth density; Small-time asymptotics; Smoothing properties of the operator Ptf (x) = E(f (X x
t ));
Functional inequalities satisfied by the law of the solution and upper Gaussian bounds for the density.
SLIDE 7 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
.
SLIDE 8 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
. Stochastic differential equations driven by fractional Brownian motions provide toy models for the study of non Markov random dynamical systems.
SLIDE 9 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
. Stochastic differential equations driven by fractional Brownian motions provide toy models for the study of non Markov random dynamical systems. Let H > 1/2. The equation is understood in the Young’s sense.
SLIDE 10 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
. Stochastic differential equations driven by fractional Brownian motions provide toy models for the study of non Markov random dynamical systems. Let H > 1/2. The equation is understood in the Young’s sense. Existence and uniqueness of the solution of (1) have been discussed by Nualart-Rascanu and Zähle.
SLIDE 11 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
. Stochastic differential equations driven by fractional Brownian motions provide toy models for the study of non Markov random dynamical systems. Let H > 1/2. The equation is understood in the Young’s sense. Existence and uniqueness of the solution of (1) have been discussed by Nualart-Rascanu and Zähle. Let H > 1/4. The equation is understood in the Lyons’ rough paths sense.
SLIDE 12 Fractional Brownian motion
A fractional Brownian motion is a Gaussian process with mean 0 and covariance function 1 2
. Stochastic differential equations driven by fractional Brownian motions provide toy models for the study of non Markov random dynamical systems. Let H > 1/2. The equation is understood in the Young’s sense. Existence and uniqueness of the solution of (1) have been discussed by Nualart-Rascanu and Zähle. Let H > 1/4. The equation is understood in the Lyons’ rough paths sense. See Coutin-Qian.
SLIDE 13
Hörmander’s type theorem
Let H > 1/2.
SLIDE 14
Hörmander’s type theorem
Let H > 1/2. If I = (i1, . . . , ik) ∈ {0, . . . , d}k, we denote by VI the Lie commutator defined by VI = [Vi1, [Vi2, . . . , [Vik−1, Vik] . . .] and d(I) = k + n(I), where n(I) is the number of 0 in the word I. Theorem (Baudoin-Hairer, PTRF’07) Assume that, at some x0 ∈ Rn, there exists N such that: span{VI(x0), d(I) ≤ N} = Rn.
SLIDE 15
Hörmander’s type theorem
Let H > 1/2. If I = (i1, . . . , ik) ∈ {0, . . . , d}k, we denote by VI the Lie commutator defined by VI = [Vi1, [Vi2, . . . , [Vik−1, Vik] . . .] and d(I) = k + n(I), where n(I) is the number of 0 in the word I. Theorem (Baudoin-Hairer, PTRF’07) Assume that, at some x0 ∈ Rn, there exists N such that: span{VI(x0), d(I) ≤ N} = Rn. (2) Then, for any t > 0, the law of the random variable X x0
t
has a smooth density with respect to the Lebesgue measure on Rn.
SLIDE 16
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting.
SLIDE 17
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting. The main difficulty is to prove a Norris type lemma for stochastic integrals with respect to fBm.
SLIDE 18
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting. The main difficulty is to prove a Norris type lemma for stochastic integrals with respect to fBm.
SLIDE 19
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting. The main difficulty is to prove a Norris type lemma for stochastic integrals with respect to fBm. Computation of the Malliavin matrix.
SLIDE 20
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting. The main difficulty is to prove a Norris type lemma for stochastic integrals with respect to fBm. Computation of the Malliavin matrix. Proof of its invertibility and Lp estimates on its inverse.
SLIDE 21
Scheme of the proof
Since Malliavin (76), the scheme of the proof is quite standard but here, it requires new estimation methods since we are in a non-semimartingale setting. The main difficulty is to prove a Norris type lemma for stochastic integrals with respect to fBm. Computation of the Malliavin matrix. Proof of its invertibility and Lp estimates on its inverse.
SLIDE 22
Recent developments
Existence of the density H > 1/4 by Cass-Friz (2010),
SLIDE 23
Recent developments
Existence of the density H > 1/4 by Cass-Friz (2010), Smoothness with H > 1/3 in some particular cases by Hu-Tindel (2011)
SLIDE 24
Recent developments
Existence of the density H > 1/4 by Cass-Friz (2010), Smoothness with H > 1/3 in some particular cases by Hu-Tindel (2011) Smoothness with H > 1/3 in the general case by Hairer-Pillai (2011) + Cass-Litterer-Lyons (2011).
SLIDE 25 Operators associated with SDEs driven by fBms
Again, let us consider the stochastic differential equations on Rn X x0
t
= x0 +
d
t Vi(X x0
s )dBi s
(3) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d dimensional fractional Brownian motion with Hurst parameter H > 1
3.
SLIDE 26 Operators associated with SDEs driven by fBms
Again, let us consider the stochastic differential equations on Rn X x0
t
= x0 +
d
t Vi(X x0
s )dBi s
(3) where the Vi’s are C ∞-bounded vector fields on Rn and B is a d dimensional fractional Brownian motion with Hurst parameter H > 1
3.
We denote by C∞
b (Rn, R) the set of compactly supported smooth
functions Rn → R. If f ∈ C∞
b (Rn, R), let us denote
Ptf (x0) = E (f (X x0
t )) , t ≥ 0,
where X x0
t
is the solution of (3) at time t.
SLIDE 27 Development in small times
Theorem (Baudoin-Coutin, SPA’07) Assume H > 1
- 3. There exists a family
- ΓH
k
- k≥0 of differential
- perators such that:
If f ∈ C∞
b (Rn, R) and x ∈ Rn, then for every N ≥ 0, when t → 0
Ptf (x) =
N
t2kH(ΓH
k f )(x) + o(t(2N+1)H).
SLIDE 28 Development in small times
1
ΓH
1 = 1
2
d
V 2
i ;
SLIDE 29 Development in small times
1
ΓH
1 = 1
2
d
V 2
i ;
2
ΓH
2 = H
4 β(2H, 2H)
d
V 2
i V 2 j +
2H − 1 8(4H − 1)
d
ViV 2
j Vi
+
4(4H − 1) − H 4 β(2H, 2H)
(ViVj)2, where β(a, b) = 1
0 xa−1(1 − x)b−1dx;
SLIDE 30 Development in small times
1
ΓH
1 = 1
2
d
V 2
i ;
2
ΓH
2 = H
4 β(2H, 2H)
d
V 2
i V 2 j +
2H − 1 8(4H − 1)
d
ViV 2
j Vi
+
4(4H − 1) − H 4 β(2H, 2H)
(ViVj)2, where β(a, b) = 1
0 xa−1(1 − x)b−1dx;
3 More generally, ΓH
k is a homogeneous polynomial in the V ′ i s of
degree 2k: ΓH
k =
aIVi1...Vi2k,
SLIDE 31
Development in small times
We conjecture that ΓH
k , k ≥ 2, has coefficients that are
meromorphic functions of H with poles in the set { 1
2j , 2 ≤ j ≤ k}.
SLIDE 32
Development in small times
We conjecture that ΓH
k , k ≥ 2, has coefficients that are
meromorphic functions of H with poles in the set { 1
2j , 2 ≤ j ≤ k}.
It would be interesting to determine what is the smallest algebra of vector fields that contains the family (ΓH
k )k≥1 ( In
the case of Brownian motion, this algebra is the algebra generated by the operator d
i=1 V 2 i ).
SLIDE 33
Smoothing property
As already seen, under the ellipticity assumption span {V1(x), · · · , Vn(x)} = Rn,
SLIDE 34
Smoothing property
As already seen, under the ellipticity assumption span {V1(x), · · · , Vn(x)} = Rn, the functional operator Ptf (x) = E (f (X x
t ))
has a smooth integral kernel.
SLIDE 35
Smoothing property
As already seen, under the ellipticity assumption span {V1(x), · · · , Vn(x)} = Rn, the functional operator Ptf (x) = E (f (X x
t ))
has a smooth integral kernel. We prove here the following regularisation bounds, for q > 1, |Vi1 · · · VikPtf (x)| ≤ Ck,q(x) tkH (Pt|f |q)1/q (x), 0 < t < 1.
SLIDE 36
Smoothing property
As already seen, under the ellipticity assumption span {V1(x), · · · , Vn(x)} = Rn, the functional operator Ptf (x) = E (f (X x
t ))
has a smooth integral kernel. We prove here the following regularisation bounds, for q > 1, |Vi1 · · · VikPtf (x)| ≤ Ck,q(x) tkH (Pt|f |q)1/q (x), 0 < t < 1. which implies |Vi1 · · · VikPtf (x)| ≤ Ck(x) tkH f ∞, 0 < t < 1.
SLIDE 37
Integration by parts on the path space
The keypoint is to use integration by parts formulas obtained through Malliavin calculus.
SLIDE 38 Integration by parts on the path space
The keypoint is to use integration by parts formulas obtained through Malliavin calculus. Since we assume ellipticity [Vi, Vj] =
n
ωk
ijVk.
SLIDE 39 Integration by parts on the path space
The keypoint is to use integration by parts formulas obtained through Malliavin calculus. Since we assume ellipticity [Vi, Vj] =
n
ωk
ijVk.
We first have the following commutation Lemma ViPtf (x) = E n
αk
i (t, x)Vkf (X x t )
where α solves the following system of SDEs: dαj
i(t, x) = n
αk
i (t, x)ωj kl(X x t )dBl t,
αj
i(0, x) = δj i .
SLIDE 40
Integration by parts on the path space
Proof. By the chain rule ViPtf (x) =
SLIDE 41
Integration by parts on the path space
Proof. By the chain rule ViPtf (x) = E ((JtVi)f (X x
t ))
SLIDE 42
Integration by parts on the path space
Proof. By the chain rule ViPtf (x) = E ((JtVi)f (X x
t )) = E ((Φt∗Vi)f (X x t )) .
SLIDE 43 Integration by parts on the path space
Proof. By the chain rule ViPtf (x) = E ((JtVi)f (X x
t )) = E ((Φt∗Vi)f (X x t )) .
Then by ellipticity, we can find αj
i(t, x) such that
Φt∗Vi(X x
t ) = n
αj
i(t, x)Vj(X x t ).
SLIDE 44 Integration by parts on the path space
Proof. By the chain rule ViPtf (x) = E ((JtVi)f (X x
t )) = E ((Φt∗Vi)f (X x t )) .
Then by ellipticity, we can find αj
i(t, x) such that
Φt∗Vi(X x
t ) = n
αj
i(t, x)Vj(X x t ).
The change of variable formula shows that α solves the above system of SDEs.
SLIDE 45
Integration by parts on the path space
We introduce the following scale of spaces:
SLIDE 46 Integration by parts on the path space
We introduce the following scale of spaces: Definition For r ∈ R, let Kr be the set of mapping Φ : (0, 1] × Rn → D∞ such that:
1 Almost surely, Φ(t, x) is smooth with respect to x and ∂Φ
∂xν is
continuous in (t, x).
SLIDE 47 Integration by parts on the path space
We introduce the following scale of spaces: Definition For r ∈ R, let Kr be the set of mapping Φ : (0, 1] × Rn → D∞ such that:
1 Almost surely, Φ(t, x) is smooth with respect to x and ∂Φ
∂xν is
continuous in (t, x).
2 For every n, p > 1,
sup
0<t≤1
t−rH
∂xν
SLIDE 48 Integration by parts on the path space
We introduce the following scale of spaces: Definition For r ∈ R, let Kr be the set of mapping Φ : (0, 1] × Rn → D∞ such that:
1 Almost surely, Φ(t, x) is smooth with respect to x and ∂Φ
∂xν is
continuous in (t, x).
2 For every n, p > 1,
sup
0<t≤1
t−rH
∂xν
SLIDE 49
Integration by parts on the path space
The key step is then the following integration by parts on the path space of fBm.
SLIDE 50
Integration by parts on the path space
The key step is then the following integration by parts on the path space of fBm. Theorem If f is C ∞-bounded and Φ ∈ Kr, E (Φ(t, x)Vif (X x
t )) = E ((TViΦ)(t, x)f (X x t ))
for some TViΦ ∈ Kr−1.
SLIDE 51 Integration by parts on the path space
Dj
sf (Xt) = ∇f (Xt), Dj sX x t
= ∇f (Xt), JtJ−1
s Vj(X x s )
=
n
hj
k(s, t, x)αk l (t, x)(Vlf )(X x t ).
where hi(s, t, x) = (βk
i (s, x)I[0,t](s))k=1,...,n,
i = 1, ..., n.
SLIDE 52 Integration by parts on the path space
Dj
sf (Xt) = ∇f (Xt), Dj sX x t
= ∇f (Xt), JtJ−1
s Vj(X x s )
=
n
hj
k(s, t, x)αk l (t, x)(Vlf )(X x t ).
where hi(s, t, x) = (βk
i (s, x)I[0,t](s))k=1,...,n,
i = 1, ..., n. Introduce Mi,j(t, x) given by Mi,j(t, x) = 1 t2H hi(·, t, x), hj(·, t, x)H. Hence Vif (X x
t ) =
1 t2H
n
βi
j(t, x)M−1 jl (t)Df (X x t ), hl(·, t)H.
SLIDE 53
Integration by parts on the path space
Using then the integration by parts formula for the Malliavin derivative, we obtain
SLIDE 54 Integration by parts on the path space
Using then the integration by parts formula for the Malliavin derivative, we obtain T ∗
ViΦ(t, x) = n
1 t2H Φ(t, x)βi
k(t, x)M−1 kl (t)δhl(·, t)
− 1 t2H D(Φ(t, x)βi
k(t, x)M−1 kl (t)), hl(·, t)H
SLIDE 55
Regularizing bounds
Iterating the previous formulas and using Hölder’s inequality, we finally conclude:
SLIDE 56
Regularizing bounds
Iterating the previous formulas and using Hölder’s inequality, we finally conclude: Theorem (Baudoin-Ouyang, 2011) For q > 1, |Vi1 · · · VikPtf (x)| ≤ Ck,q(x) tkH (Pt|f |q)1/q (x), 0 < t < 1.
SLIDE 57
Global bounds
SLIDE 58
Global bounds
In some geometric situations, it is possible to obtain global bounds independent from x.
SLIDE 59
Global bounds
In some geometric situations, it is possible to obtain global bounds independent from x. Assume the skew-symmetry condition ωk
ij = −ωj ik,
SLIDE 60 Global bounds
In some geometric situations, it is possible to obtain global bounds independent from x. Assume the skew-symmetry condition ωk
ij = −ωj ik,
then we have the global bound
(ViPtf )2(x) ≤ Pt
(Vif )2 (x).
SLIDE 61
Global bounds
In a recent work with C. Ouyang and S. Tindel, we proved that under the same structure assumptions, we have the Gaussian concentration
SLIDE 62 Global bounds
In a recent work with C. Ouyang and S. Tindel, we proved that under the same structure assumptions, we have the Gaussian concentration Theorem (Baudoin-Ouyang-Tindel, 2011) There exists M such that for every T ≥ 0 and λ ≥ 0, P
0≤t≤T
X x
t − E
0≤t≤T
X x
t
λ2 2MT 2H
SLIDE 63 Global bounds
In a recent work with C. Ouyang and S. Tindel, we proved that under the same structure assumptions, we have the Gaussian concentration Theorem (Baudoin-Ouyang-Tindel, 2011) There exists M such that for every T ≥ 0 and λ ≥ 0, P
0≤t≤T
X x
t − E
0≤t≤T
X x
t
λ2 2MT 2H
and a corresponding Gaussian upper bound.
SLIDE 64
Global bounds
Theorem (Baudoin-Ouyang-Tindel, 2011) For any t ∈ R∗
+, the random variable X x t admits a smooth density
pX(t, ·).
SLIDE 65 Global bounds
Theorem (Baudoin-Ouyang-Tindel, 2011) For any t ∈ R∗
+, the random variable X x t admits a smooth density
pX(t, ·). Furthermore, there exist 3 positive constants c(1)
t
, c(2)
t
, c(3)
t,x such that
pX(t, y) ≤ c(1)
t
exp
t
t,x
2 , for any y ∈ Rd.
SLIDE 66
Global bounds
Again, under the structure assumption we also have a global Poincaré inequality:
SLIDE 67 Global bounds
Again, under the structure assumption we also have a global Poincaré inequality: Theorem (Baudoin-Ouyang-Tindel, 2011) Pt(f 2) − (Ptf )2 ≤ Ct2HPt n
(Vif )2
SLIDE 68 Global bounds
Again, under the structure assumption we also have a global Poincaré inequality: Theorem (Baudoin-Ouyang-Tindel, 2011) Pt(f 2) − (Ptf )2 ≤ Ct2HPt n
(Vif )2
- a log-Sobolev inequality:
Theorem (Baudoin-Ouyang-Tindel, 2011) Pt(f ln f ) − (Ptf )(ln Ptf ) ≤ 2Ct2HPt n
(Vi ln f )2 f
